Mikhail (Mike, Misha) Kagan jointly with Misha Klin Resistance-distance transform (RDT) in the context of Weisfeiler-Leman stabilization (WLS) Mikhail (Mike, Misha) Kagan jointly with Misha Klin WL 2018

Outline Main idea of the approach RDT in a nutshell Different approaches to RDT computations Teaching example: WLS vs RDT Some computations and observations Conclusions Further goals References

Main idea of the approach We start with a simple colored graph on n vertices Using Kirchhoff’s rules (Kirchhoff, 1847), we compute so-called resistance distances (RD) between every pair of vertices We stratify (now) complete graph Kn according to diverse values of RD’s We obtain a coloring of Kn, which is considered as imitation of Weisfeiler-Leman stabilization (Weisfeiler, 1976)

RDT in a nutshell. Reminder

RDT in a nutshell. Examples Rab = 2 a b dab = 2 Rab = 1 a b dab = 2 Rab = 2/3 a b dab = 2

RDT in a nutshell. General definition of RD Given electric network on n vertices with edge resistances Rij Edge conductance ⟶ weighted adjacency matrix A Complete colored graph Г with colors σij E.g. simple graph: edge ⟷ σij = 1, non-edge ⟷ σij = 0 RD (i , j) = effective resistance b/w vertices i and j , provided ideal battery connected across i and j Resistance Distance Transform A ⟶ Ã = new weighted adjacency matrix with entrees 0 if i = j and 1/RD(i, j) otherwise

RDT in a nutshell. RD as a metric Resistance distance satisfies (Klein & Randič, 1993) R (a, b) ≥ 0 R (a, b) = 0, iff a=b R (a, b) = R (b, a) R (a, b) ≤ R (a, c) + R (c, b) For a tree: RD = standard distance The symmetry of RD will be of one the reasons for limitation of the approach

Different approaches to RD computations Provided Adjacency matrix: Aij = Aji Degree matrix: Dij = diag (d1, d2, …), di = ΣjAij Laplacian matrix: L = D - A Resistance Distance Method 1 (Klein & Randič, 1993) where Гij = Moore-Penrose pseudo inverse of Laplacian matrix Method 2 (Bapat, 1999; MK, 2015)

Different approaches to RD computations Method 3 (Bapat & Gupta, 2010) 2-forest separating i and j = forest with n – 2 edges and two connected components, such that i and j belong to different components Example 1 2 3 1 2 3 1 2 3 1 2 3 One tree One 2-forest for 1 & 2 Two 2-forests for 1 & 3 RD (1,2) = RD (2,3) =1 RD (1,3) = 2

Teaching Example: WLS vs RDT WLS procedure Non-commutative variables: a for each vertex b for each edge c for each non-edge Replace distinct polynomials with new distinct variables Repeat until stabilization: Mk+1 ≈ Mk . In this example k = 2.

Teaching Example: WLS vs RDT RDT procedure Need to compute: RD (1,2) RD (1,3) RD (2,4) New Laplacian matrix (RDT of L)

Teaching Example: WLS vs RDT Comparison of WLS and RDT results L2 = symmetrized version of M2

Some computations and observations Distance regular graphs RDT was performed for a number of connected DRG’s (Г). In each case, the number of distinct colors was equal to the diameter d of Г. In addition, obtained ”patterns” of RD’s for algebraically isomorphic, but not isomorphic DGR’s, were also equivalent. The computations were performed for : Shrikhande and L2(4) on 16 vertices 15 and 10 Paulus graphs on 25 and 26 vertices Chang graphs on 28 vertices Dodecahedron (d=5) on 20 vertices

Some computations and observations Two Brouwer’s {0,2}-graphs Two {0, 2}-graphs (in the sense of Andries Brouwer) on 20 vertices were considered: The vertex-transitive N6.3 and The N6.4 with Aut of order 24 In both cases, the result of symmetrization of the WL-closure coincided with RDT For N6.3, RDT yields Jordan closure (in the sense of P. Cameron)

Some computations and observations Co-spectral example: 4-Cube and Hoffman For the 4-Cube (Q4) of diameter 4, standards results are obtained: RDT = WLS For Hoffman graph, which is co-spectral to Q4, the WL-closure has rank 19, with 3 fibers. Its symmetrization has rank 11. The first iteration of RDT yields 2 fibers with 6 colors. The result of the second iteration of RDT coincides with the symmetrization of WL closure: 3 fibers and 11 colors

Some computations and observations Two Circulant graphs on 20 vertices There are two non-isomorphic co-spectral circulants on 20 vertices, both generate the same coherent closure We get the same “patterns” applying RDT, however the two graphs can be distinguished by their distinct sets of RD values

Conclusions RDT provides an intriguing approach to imitate WLS Comparison with WLS: For certain graphs (e.g. DRG’s) RDT leads to the same result (corollary of theorems by Norman Biggs [3]) RDT often stabilizes in one iteration, provided that minimal degree (valency) is at least 2 Limitations In considered examples, RDT provides symmetrized version of WLS Sometimes requires more than one iteration (Hoffman)

Future goals To perform more experiments and to interpret them To prove rigorously that RDT(Г ) is a part of WL(Г ) To understand when RDT is a Jordan scheme To generalize the results of Biggs to AS’s

References [1]  R. Bapat, “Resistance distance in graphs,’’ Mathematics Student, Vol 68, Nos. 1-4 (1999) [2]  R. Bapat and S. Gupta, “Resistance distance in wheels and fans,’’ Indian J. Pure Appl. Math, 41(1), 1-13 (2010) [3] N. Biggs, “Potential Theory on Distance-Regular Graphs”, Combinatorics, Probability and Computing, 2(3), 243-255 (1993) [4]  M. Kagan, “On equivalent resistance of electrical circuits,’’ Am. J. Phys. 83, 53-63 (2015) [5] G. Kirchhoff, “Über die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Strömegeführt wird,” Ann. Phys. Chem. 148, 497–508 (1847) [6]  D. J. Klein and M. Randič, “Resistance distance,’’ J. Math. Chem., Vol 12, 81-95 (1993) [7] B. Weisfeiler (editor), “On construction and identification of graphs’’. Lecture Notes in Mathematics, Vol. 558. Springer-Verlag, Berlin- New York, 1976. xiv+237 pp.

Thank you for your attention! RDT = symmetrized version of WLS